The class blog for Math 3010, fall 2014, at the University of Utah

Tag Archives: zeno’s paradox

Our knowledge of mathematics develops along with the long history of human civilization. Ancient Greece is usually considered as the cradle of western civilization and the birthplace of mathematics. Here I will discuss the famous Zeno’s Paradox, an intellectual legacy we inherited form those great thinkers in ancient Greece, whose philosophical thinking has been energetic and attractive since ancient times; Then I will have a brief introduction about the “solving” of the paradox using geometric series; In the end I will show that, in some sense, the paradox has not been fully unraveled, by reference to another problem proposed by contemporary scholars. I believe the charm of mathematics will be presented after these efforts.

The ancient Greek philosopher Zeno once created quite a few paradoxes to show his skepticism about some common phenomena. He thought plurality and change were not a universal truth, and in particular, motion was only our illusion. Among his paradoxes that survived today, most of them have equivalent math models. So I will pick up one of them, “Achilles and the Tortoise”, to represent his logic.

The problem is like this: Achilles, the most famous Achaean warrior in Homer’s Iliad, the “swift-footed” hero, is chasing a tortoise. Suppose the initial distance between them is 100-meters, and Achilles’ speed is 10 m/s while the tortoise’s speed is 1m/s. After the chasing begins, Achilles will spend 10 seconds to finish a first the 100-meters. Then he will be at the spot where the tortoise was, at 10 seconds ago; In this period (10 seconds), the tortoise also proceeds 10 meters. Then, to finish the second distance, 10 meters, Achilles spends 1 second, while in the same period, the tortoise proceeds 1 meter; Then it goes on, every time Achilles reaches the tortoise’s previous spot, he still needs to chase more because in that period the tortoise proceeds to another further spot. Hence, Zeno concludes, in this case, Achilles will never overrun the lucky tortoise, which is a very bizarre conclusion against our common sense.

This paradox raised in history of great interest. Many scholars tried to give an answer or explanation, including Aristotle, Archimedes, Thomas Aquinas, etc. The joint efforts of philosophers and mathematicians did not succeed immediately. Without the help of rigorous mathematical tools, their solutions cannot resist questioning from skepticism. To make it more clear, philosophical thinking alone could hardly solve this problem; even if it accomplished so, to convince others to believe this will be no less difficult. Immanuel Kant in his Critique of Pure Reason mentioned that rationality is not omnipotent. It has its own structure of a priori knowledge, and after itself combined with a posterior experience, it becomes useful knowledge, which guides our cognition. However, due to the nature of human’s longing for perfection, eternal, and universality (I would like to add “infinite” here), we are inclined to abuse our rationality and expands it to areas that it in fact does not apply. This is to say, human rationality arises from very specific experience, and is applicable there; but due to our preference, we create some concepts (like “perfection”, “eternal” and “universality” I mentioned above), which is non-existent in real life and also beyond rationality’s realm. But we are so confident and accustomed to our rationality that we apply it to those concepts generated by ourselves, without noticing it is not applicable there. After the abuse, confusion subsequently follows.

I really admire Kant’s genius in his noticing that a critique of human reasoning itself is very much needed. And I would use his theory to help form my personal understanding about this problem. But I will leave it here and deal with it later, after the introduction of the rigorous mathematical proving with respect to this problem.

Thanks to the invention of calculus and the epsilon-delta language, we now have the rigorous mathematical tool to deal with problems about infinity. A brief solution is to use geometric series. With respect to the “Achilles and Tortoise” problem we mentioned above, the time that Achilles needed to catch up with the tortoise can be represented as:

This means Achilles could overrun the tortoise after approximately 11.11 seconds. Thus, the sum of a series with infinite terms, are quite possibly finite, which may be beyond our predecessors’ understanding. But, does this problem stops here? Some modern scholars believes not. Why, because we are not sure what is Zeno’s true meaning. This is to say, the result of the formula may not answer Zeno’s question. Let me here give an example, which is called Thomson’s Lamp: suppose there is such a lamp with a toggle switch. After you start the game, it’s switched one after 1 minute, then switched off after half minute, then on after fourth minute, then off after eighth minute, and so goes on. The sum all the time we spend in the game is 2 minutes, according to the same method about sum of geometric series above. Then, one question follows: After exactly two minutes, is the lamp on or off?

This time we find it’s also very difficult to answer this variation of Zeno’s paradox, even if we know geometric series. And because of this, I believe to use geometric series could give a result, but could not solve the problem about the process, which may be Zeno’s real point. And Kant’s argument gives me guidance in understanding this paradox. Also, there is a scholar making this point more explicitly: according to Pat Corvini, this paradox arises from “a subtle but fatal switch between the physical and abstract”. When we expand our mathematical abstractions to the physical world, even it’s applicable almost everywhere, with respect to some concepts, it’s quite unimaginable and confusing. This time, we may still need mathematics as well as philosophy, to finally solve this paradox.

No person can create a true infinity. There is no number associated with it, only an idea of never-ending numbers. Therefore, logic says a bounded line would contain a finite number of points rather than an infinite number. Zeno decided that this was not accurate, in that there is always a half-way point on any bounded line, no matter how many times you chop it up.

Zeno (or Xeno) of Elea was a Greek philosopher of the Eleatic School, easily best known for his paradoxical arguments having to do with time and space and the impossibility of motion in any logical sense. Mentioned in writings of both Plato and Aristotle, he showed that viewing space as a multitude of points and time as a multitude of “moments” concludes that motion is an illusion. But thanks to common sense we can dispute that through our own sensory perception, motion is indeed a thing (though some philosophers would love to argue that our senses are merely deceiving us on a number of levels). Nevertheless, Zeno proposed four arguments against motion, among them The Dichotomy.

The Dichotomy is a very similar argument to the bisecting line mentioned before. Zeno’s example describes a horse attempting to cover the distance between points A and B. Now, in order for the horse to reach its end point, it must first reach a midpoint, right? Of course. But in order to reach that midpoint, it must reach a quarter point, and so on. Therefore, if space truly contains an infinite set of points, the horse must cross an infinite number of these points in a finite amount of time. This statement is paradoxical, so by this reasoning the horse will never reach point B, because it cannot even move from point A.

As mentioned before, infinity is merely a concept. But it is a concept that mathematicians have, if begrudgingly, come to accept and even apply. So thanks to modern calculus, we do have a mathematical solution to Zeno’s The Dichotomy: the distance the horse is to cross can be explained as a series such as ½ + ¼ + 1/8 + … When it comes to a series, an easy way to solve this one in particular is by using partial sums. With partial sums, one could recognize that the sequence, otherwise expressed as 1/2 + 1/(2^2) + 1/(2^3) + …, follows a specific pattern that results in the sum after n terms being 1-(1/2^n). Bringing in limits and the concept of infinity, and assuming that the horse runs at a fairly constant speed, we can conclude that these sums become increasingly close to 1. In fact, the series converges to 1, meaning the horse does not require an infinite amount of time to complete its task and movement is indeed possible.

Because of its useful application in problems such as The Dichotomy, infinity has become a well-used and helpful tool, so long as you don’t look too closely at the actual implication behind the concept. To say an infinite set of numbers will “eventually” reach a finite number seems like such a preposterous conclusion to draw, but alas we can. Physicists will also chime in on such a paradox as this with the fact that if you look close enough, you will not find infinite space, but a finite, albeit incomprehensibly large, set of atoms and other such building blocks of nature. These microscopic materials do indeed seem to be Mother Nature’s way of saying infinity is a silly concept in real world application, though the term is still used in regular calculations throughout physics and engineering. As my calculus teacher is fond of mentioning, Engineers and Physicists will look at an application of infinity such as the one used to solve The Dichotomy, shrug, and say “it’s close enough”. As mathematicians, we may nod in agreement, but we know better.